Consider an ODE of the form $$y''=f(x,y,y') $$ with $x \geq 0$, and initial conditions of the form $$y(0)=y_0>0, \\y'(0)=m_0. $$ I want to claim that under reasonable conditions (which I can't precisely formulate at the moment) on the function $f$, we have the following
The solution $y$ with the initial conditions above will attain a negative value in its interval of existence, provided that $m_0$ is sufficiently large and negative.
I was wondering if such an existence result for boundary value problems is known. If so, I'd like to see a reference. Otherwise, I'd like help in formulating the statement properly.
Intuitively, the solution $y(x)$ behaves similarly to $y_0+m_0 x$ for $x$ near $0$, so taking $m_0$ large and negative should result in an intersection with the $x$-axis. However, I can't see why the interval of existence doesn't shrink in size as $m_0$ decreases, preventing such an intersection via a singularity.
Thanks in any case!
